Find the LCM of fractions clearly What is the least common multiple (LCM) of the fractions 2/3, 4/9, 5/6, and 7/12? Use the standard rule: for positive fractions, LCM = LCM of numerators / HCF of denominators.

Introduction / Context:
This problem asks for the least common multiple (LCM) of several fractions. For fractions, there is a well-known formula that converts the multi-step process into two simpler integer tasks: take the LCM of all numerators and divide it by the highest common factor (HCF or GCD) of all denominators.

Given Data / Assumptions:

• Fractions: 2/3, 4/9, 5/6, 7/12.
• All numbers are positive.
• Rule: LCM of fractions = LCM(numerators) / HCF(denominators).

Concept / Approach:
Compute two values independently: (1) LCM of numerators and (2) HCF of denominators. Then divide the first by the second. This avoids converting each fraction to many equivalent forms.

Step-by-Step Solution:
Numerators: 2, 4, 5, 7 ⇒ LCM = 140.Denominators: 3, 9, 6, 12 ⇒ HCF = gcd(3, 9, 6, 12) = 3.Therefore LCM of the fractions = 140 / 3.

Verification / Alternative check:
Confirm 140/3 is a common multiple: (140/3) ÷ (2/3) = 70, (140/3) ÷ (4/9) = 105, (140/3) ÷ (5/6) = 56, (140/3) ÷ (7/12) = 80. All are integers; trying any smaller candidate fails for at least one fraction.

Why Other Options Are Wrong:
1/18 and 1/36 are far too small. 35/9 is not a common multiple for all given fractions. 28/3 is a common multiple for some but not the least for all.

Common Pitfalls:
Mixing up the rule (using HCF of numerators / LCM of denominators, which is actually for the HCF of fractions), or computing LCM and HCF pairwise rather than across the full set.

Final Answer:
140/3